The well-known argument, which allows to conclude that the existence of a density \(f_N\)  for some \(N\in \mathbb {N}\)  implies the existence of densities \(f_n\)  for any \(n\ge N\)  which holds for the CLT-statistic \(Z_n\)  does not apply here.
For both statistics \(T_n\)  and \(Z_n\) , conditions of this type do not imply that \(X_1\)  is absolutely continuous or even has a non-zero absolutely continuous component. For a counterexample with \(X_1,X_2\)  having singular continuous distribution and \(X_1+X_2\)  being absolutely continuous with bounded density, see [1]}.
